Threshold phenomena for high-dimensional random polytopes

Let \(X_1,\ldots,X_N\), \(N>n\), be independent random points in \(\mathbb{R}^n\), distributed according to the so-called beta or beta-prime distribution, respectively. We establish threshold phenomena for the volume, intrinsic volumes, or more general measures of the convex hulls of these random...

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Bibliographic Details
Published inarXiv.org
Main Authors Bonnet, Gilles, Chasapis, Giorgos, Grote, Julian, Temesvari, Daniel, Turchi, Nicola
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 14.06.2018
Subjects
Convexity
Hulls
Mathematics - Metric Geometry
Mathematics - Probability
Polytopes
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Summary:Let \(X_1,\ldots,X_N\), \(N>n\), be independent random points in \(\mathbb{R}^n\), distributed according to the so-called beta or beta-prime distribution, respectively. We establish threshold phenomena for the volume, intrinsic volumes, or more general measures of the convex hulls of these random point sets, as the space dimension \(n\) tends to infinity. The dual setting of polytopes generated by random halfspaces is also investigated.
ISSN:2331-8422
DOI:10.48550/arxiv.1802.04089